Labyrinth Patterns Research Articles

Phase-field modeling of peritectic coupled growth in the TRIS–NPG model system

Microstructures forming during the directional solidification of the hypo-peritectic TRIS–NPG model alloy were studied by phase-field simulations. Steady state growth forms could be obtained under conditions similar to those in recent space experiments. In two dimensions, the wavelength range of stable lamellar structures has been determined as function of the temperature gradient and the melt composition. Front temperatures extracted from simulations were compared to the predictions of the modified Jackson–Hunt theory. In three dimensions, structures and phenomena known from eutectic systems were reproduced. Transitions between lamellar and rod structures were induced by slowly changing the melt composition. The regularizing effect of tilting the temperature gradient, transforming a random labyrinth pattern to an ordered one, has also been demonstrated. These simulations show the universality of the basic governing mechanism — that is, the competition of solute diffusion with capillary effects — in the pattern formation of these multi-phase systems.

Spatiotemporal Patterns in a Space–Time Discrete SIRS Epidemic Model with Self- and Cross-Diffusion

In this work, a two-dimensional Coupled Map Lattice (CML) method is applied to a continuous Reaction–Diffusion (RD) SIRS epidemic model. The model involves a complex, nonlinear incidence rate and the spatially heterogeneous self- and cross-diffusion coefficients. The CML method applied here results in a space–time discrete counterpart of the model. The basic reproduction number is derived, and the local stability of the fixed points of the discrete model is established. The two typical bifurcations of codimension 1, such as the flip and the Neimark–Sacker (NS) bifurcations, are investigated by utilizing the center manifold theory and the normal form theorem. Furthermore, the codimension-2 fold–flip bifurcation about the endemic fixed point is discussed in detail. Turing instability criterion of the cross-diffusion epidemic model is established by utilizing the CML method under periodic boundary conditions. Finally, the numerical simulations clarify both the effectiveness of all theoretical findings and the effects of time step size and cross-diffusion on the spatiotemporal patterns. The main contribution of this work is the discovery of some interesting Turing and non-Turing patterns which may be induced by flip–Turing instability, NS–Turing instability or fold–flip–Turing instability. Concretely speaking, spiral patterns are observed near the flip bifurcation threshold value and ultimately evolve into an irregular defect pattern. Stripe pattern, labyrinth pattern and circular pattern are observed near the NS bifurcation threshold value. The disorderly interwoven ribbon pattern, mosaic pattern and some irregular oscillatory patterns are observed near the fold–flip bifurcation point. Our results greatly help in predicting the long-term behavior of the space–time discrete SIRS epidemic model under study.

Phase coexistence and transitions between antiferromagnetic and ferromagnetic states in a synthetic antiferromagnet

In synthetic antiferromagnets (SAFs), antiferromagnetic (AFM) order and synthesis using conventional sputtering techniques is combined to produce systems that are advantageous for spintronics applications. Here we present the preparation and study of SAF multilayers possessing both perpendicular magnetic anisotropy and the Dzyaloshinskii-Moriya interaction. The multilayers have an antiferromagnetically aligned ground state but can be forced into a full ferromagnetic (FM) alignment by applying an out-of-plane field ∼100 mT. We study the spin textures in these multilayers in their ground state as well as around the transition point between the AFM and FM states at fields ∼ 40 mT by imaging the spin textures using complementary methods: photoemission electron, magnetic force, and Lorentz transmission electron microscopies. The transformation into a FM state by field proceeds by a nucleation and growth process, where skyrmionic nuclei form and then broaden into regions containing a ferromagnetically aligned labyrinth pattern that eventually occupies the whole film. Remarkably, this process occurs without any significant change in the net magnetic moment of the multilayer. The mix of antiferromagnetically and ferromagnetically aligned regions on the micron scale in the middle of this transition is reminiscent of a first-order phase transition that exhibits phase coexistence. These results are important for guiding the design of spintronic devices whose operation is based on spin textures in perpendicularly magnetized SAFs. Published by the American Physical Society 2024

Exploring the impact of spatial patterns on restoration efforts: promoting self‐facilitating feedback mechanisms with an innovative biodegradable seed mussel collector

Transplantations of organisms in aquatic ecosystems play an important role in ecological restoration and commercial practices. However, success rates of these transplantations, especially when ecosystem engineers are involved, are often low. To enhance transplantation success, the promotion of self‐facilitation between transplants that mitigate environmental stressors is crucial. Besides, spatial patterns resulting from self‐facilitation can enhance ecosystem resilience. Using blue mussels as a model organism, we explored the possibility of increasing transplantation success in a subtidal ecosystem. We used biodegradable structures (“BioShell‐SMCs”) to ameliorate self‐facilitating feedback mechanisms to overcome environmental stressors in the initial post‐transplantation phase, and to increase transplantation success by implementing large‐scale spatial configurations, mimicking natural mussel bed patterns. The structures are an innovation of traditional seed mussel collectors (SMCs) used in mussel cultivation. They consist of a biodegradable net based on a compound of aliphatic polyesters, filled with empty cockle shells around a coconut fiber rope. We tested whether different spatial configurations could increase transplantation success of mussel seed: low versus high density labyrinth pattern and banded pattern. The results of this experiment showed high losses (approximately 75%), with no significant variation between configurations. The lack of migration due to unexpected retention of the biodegradable net hindered the initiation of natural aggregations, resulting in increased competition among mussels. Besides, factors such as hydrodynamic dislodgement, burial and interannual variation likely contributed to the observed losses. While the BioShell‐SMC has not demonstrated large‐scale success, this research contributes to understanding the mechanisms that underlie successful transplantation strategies in aquatic ecosystems.

Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect

Cooperation during hunting is a common predation plan of action among several large predators to increase their biomass raising capturing potential which can also produce fear in the prey and dispersal-influenced pattern creation is also relevant from both a fundamental and an applied standpoint. Considering these, we present a modified Leslie–Gower predator–prey model incorporating fear, intra-specific competition and hunting cooperation of predators following Berec’s encounter-driven functional response in presence of cross-diffusion. In the non-spatial system, biologically feasible equilibria and their stability conditions are derived. Taking the level of fear as bifurcation parameter, occurrence of limit cycle via Hopf bifurcation and the stability of limit cycle computing Lyapunov coefficient are discussed. In diffusive model, spacial focus on Turing instability conditions and to determine the relevant intervals at which various patterns will begin to appear. In the two dimensional spatial domain, under Neumann boundary condition, various Turing patterns such as spot, stripe, mixture of spot and stripe, labyrinth patterns are emerged varying fear, hunting cooperation and cross-diffusion coefficient. Hopf–Turing region is also a centre for several complex patterns. The numerical simulations indicate that the patterns are in same or opposite phase according to the sign of cross-diffusion coefficient. Intra-specific competition among predators tend the system towards stability. Cross-diffusion causes a paradox to population density with raising fear and hunting cooperation.

Labyrinth pattern in a decagonal quasicrystal

Labyrinth pattern in a decagonal quasicrystal

3D-Printed Bioreceptive Tiles of Reaction–Diffusion (Gierer–Meinhardt Model) for Multi-Scale Algal Strains’ Passive Immobilization

The current architecture practice is shifting towards Green Solutions designed, produced, and operated domestically in a self-sufficient decentralized fashion, following the UN sustainability goals. The current study proposes 3D-printed bioreceptive tiles for the passive immobilization of multi-scale-length algal strains from a mixed culture of Mougeotia sp., Oedogonium foveolatum, Zygnema sp., Microspora sp., Spirogyra sp., and Pyrocystis fusiformis. This customized passive immobilization of the chosen algal strains is designed to achieve bioremediation-integrated solutions in architectural applications. The two bioreceptive tiles following the reaction-diffusion, activator-inhibitor Grier–Meinhardt model have different patterns: P1: Polar periodic, and P2: Strip labyrinth, with niche sizes of 3000 µm and 500 µm, respectively. The results revealed that P2 has a higher immobilization capacity for the various strains, particularly Microspora sp., achieving a growth rate 1.65% higher than its activated culture density compared to a 1.08% growth rate on P1, followed by P. fusiformis with 1.53% on P2 and 1.3% on P1. These results prove the correspondence between the scale and morphology of the strip labyrinth pattern of P2 and the unbranched filamentous and fusiform large unicellular morphology of the immobilized algal strains cells, with an optimum ratio of 0.05% to 0.75% niche to the cell scale. Furthermore, The Mixed Culture method offered an intertwining net that facilitated the entrapment of the various algal strains into the bioreceptive tile.

The impact of heterogeneous human activity on vegetation patterns in arid environments

Vegetation patterns have attracted increasing interest in recent years since they can be used as a key indicator of ecosystem robustness. As one of the vital factors affecting vegetation structures, human activities have been widely explored in the literature. Nevertheless, the effects of spatiotemporal heterogeneity of human activities on vegetation patterns are far from being well explained. Here, we address this issue by applying optimal control theory to the dryland vegetation-water model. We find that the spatiotemporal heterogeneity of human activities leads to the transition from different states to desired vegetation patterns, including the spot, labyrinth, and gap patterns, and thus increases the diversity of pattern structures. The heterogeneity in human activities is also found to promote the vegetation growth in low-rainfall areas, which in turn effectively prevents vegetation desertification. Our robustness analysis fully supports these findings. This work well quantitatively links human activities with ecosystems robustness and helps provide new insights for biodiversity conservation.

Vortex structure in relaxed BaTiO3/SrTiO3 superlattice

In this work, using the phase-field model, we demonstrated vortex structure can be generated in a lead-free BaTiO3/SrTiO3 superlattice structure. The labyrinth pattern (maze-like) domain structure was observed in a fully relaxed BaTiO3/SrTiO3 superlattice structure. Topological bubble-like rosette structures were found at an out-of-plane field, and polar skyrmions were obtained after the electric field was removed. In a designed fully relaxed nanowire of width ∼8 nm, near-perfect periodical clockwise-counterclockwise ferroelectric vortex pairs were generated due to the interfacial constraints. The simulated topological structures are potentially relevant to the application of high-density ferroelectrics memory devices.

Impacts of prey-taxis and nonconstant mortality on a spatiotemporal predator–prey system

Prey-taxis, which plays an important role in biological control and ecological balance, is the phenomenon that predators shift directly toward regions with the highest density of prey. In this paper, we consider a spatiotemporal predator–prey system involving prey-taxis and nonconstant mortality. By structuring Lyapunov functional, we obtain the global stability of spatially homogeneous steady states. The minimal mortality of predator will determine whether they can survive. In addition, we identify the parameter ranges that spatially homogeneous steady state remains stable or becomes unstable, and discuss steady state bifurcation by choosing prey-taxis coefficient. It is found that a diffusion-induced instability can occur under nonconstant mortality, and prey-taxis can annihilate the spatial patterns induced by diffusion. Numerical simulations illustrate that prey-taxis and nonconstant mortality can result in rich pattern formations including spots pattern, spots–strips pattern, labyrinth pattern, strips-like pattern. It is worth noting that the weak prey-taxis is disadvantageous in terms of the number of two species.

Gabriel Faivre: his legacy on eutectic solidification

In the honor of Gabriel Faivre (1944-2020), I will present a review of major scientific contributions to the understanding of the dynamics of eutectic growth patterns. From the end of the 1980s, Gabriel Faivre undertook a systematic research in solidification guided by the new concepts of the nonlinear physics of out-of-equilibrium pattern formation. Drawing on his outstanding capabilities as an experimentalist, he refined the method of in situ directional solidification of model alloys. With constant reference to physics and metallurgy, he succeeded in carrying out a high-level research, keen to reach strong qualitative impact and quantitative accuracy. Gabriel Faivre made key discoveries, together with coworkers and young researchers in Paris, and in collaboration with materials scientists and physicists in France and abroad. From symmetry breaking instabilities to eutectic cells and dendrites, over rod-like and labyrinth patterns, full light has been shed onto new phenomena, fascinating to the eye and the mind. During the last decade, Gabriel Faivre mentored an in-depth analysis of interfacial-anisotropy effects on coupled-growth patterns, thus reconciliating the theories of regular eutectics and crystal-orientation dependent eutectic-grain growth. Being both a rigorous scientist and a generous colleague, he left us a vast legacy of prospective research topics in solidification and crystal-growth science. Sharing his knowledge of fine arts and humanities, Gabriel Faivre also instilled the best of intellectual thinking in those who were fortunate enough to work with him.

Spatiotemporal dynamics and Turing patterns in an eco-epidemiological model with cannibalism

In this work, we have studied the spatiotemporal dynamics and pattern formation in an eco-epidemiological model with cannibalism, which are not explored in the existing literature. Cannibalism is a process of killing and consuming the full or a part of an individual of the same species (conspecifics). A cannibalistic reaction–diffusion system is proposed and analyzed. In order to observe the change in system stability, we have investigated local and global asymptotic stability for the constant positive steady state. The existence of Hopf bifurcation and Turing instability are proved to understand the change in the system dynamics and to observe the species movement. Numerical simulation is performed to observe spatiotemporal dynamics and formation of Turing patterns. The existence of Hopf bifurcation is verified numerically in the spatiotemporal system It is observed that cannibalism stabilizes the temporal system, which is shown by bifurcation diagrams, phase portraits and time-series plots. Moreover, its increased cost promotes the occurrence of Turing patterns in the spatiotemporal system (i.e., checked numerically that for an increased cost of cannibalism, Turing instability holds). The movement of prey population also plays an important role in forming Turing patterns, as a significant change in movement results in various types of interesting two-dimensional Turing patterns such as hot-spot, stripe and labyrinth patterns. We see that both predator species show the same species distribution arrangements under cannibalistic pressure. Hence cannibalism can be seen as an essential factor for the significant change in spatiotemporal dynamics and species distribution arrangement.

Spatial dynamics of a vegetation model with uptake-diffusion feedback in an arid environment.

Vegetation patterns with a variety of structures is amazing phenomena in arid or semi-arid areas, which can identify the evolution law of vegetation and are typical signals of ecosystem functions. Many achievements have been made in this respect, yet the mechanisms of uptake-diffusion feedback on the pattern structures of vegetation is not fully understood. To well reveal the influences of parameters perturbation on the pattern formation of vegetation, we give a comprehensive analysis on a vegetation-water model in the forms of reaction-diffusion equation which is posed by Zelnik et al.(Proc Natl Acad Sci 112:12,327-12,331, 2015). We obtain the exact parameters range for stationary patterns and show the dynamical behaviors near the bifurcation point based on nonlinear analysis. It is found that the model has the properties of spot, labyrinth and gap patterns. Moreover, water diffusion rate prohibits the growth of vegetation while shading parameter promotes the increase of vegetation biomass. Our results show that gradual transitions from uniform state to gap pattern can occur for suitable value of parameters which may induce the emergence of desertification.

ON MIXED TRIANGULAR LABYRINTHIC FRACTALS

We introduce and study mixed triangular labyrinthic fractals, which can be seen as an extension of (generalized) Sierpiński gaskets. This is a new class of fractal dendrites that generalize the self-similar triangular labyrinth fractals studied recently by the authors. A mixed triangular labyrinthic fractal is defined by a triangular labyrinthic pattern and an infinite sequence of triangular labyrinth patterns systems, whereas one triangular labyrinth patterns system is sufficient for generating a self-similar triangular labyrinth fractal. Moreover, a labyrinthic pattern is more general than a labyrinth pattern, and the idea is that the use of many different patterns provides objects with a “richer” structure. After proving that these non-self-similar fractals are dendrites, we study the growth of path lengths in graphs associated with iterations. We prove geometric properties of the fractals, some of which have rather “local” character and on the other hand occur at sites distributed “all over” the fractals.

Kirigami-inspired adhesion with high directional asymmetry

Robust adhesion to and effortless removal from various substrates are of great importance to soft adhesives in myriad applications. However, controlled regulation between strong attachment and easy detachment in a same system remains challenging. Herein, we report a kirigami-inspired approach to control adhesion with high directional asymmetry. By inscribing a specially designed labyrinth pattern in an adhesive film, the adhesion is enhanced against peeling along a particular direction – the peak peel force builds up at a much smaller peeling angle, and the adhesion energy climbs through in-plane elastic dissipaters. Such a mechanism is inactive when the adhesive is peeled in the opposite direction, and easy detachment is experienced. The underlying physics is elucidated via theoretical and numerical analyses and verified through experiments by using hydrogel adhesives with different elastic moduli and geometric parameters. The asymmetric adhesion can be further enhanced over soft substrates or by designing hierarchical structures – increases by up to 22 folds in strength and 7 folds in energy have been observed. The proposed mechanism is universally applicable to various intermolecular interactions and different loading modes, and can pave the way to engineering new adhesives with superior yet controllable adhesion performance.

Impact of environmental asymmetry on epithelial morphogenesis

Epithelial folding is a universal biological phenomenon in morphogenesis, typical examples being brain gyri, villi of the intestinal tract, and imaginal discs in invertebrates. During epithelial morphogenesis, the physical constraints imposed by the surrounding microenvironment on epithelial tissue play critical roles in folding morphology. In this study, we focused on the asymmetry of the environmental constraints sandwiching the epithelial sheet and introduced the degree of asymmetry, which indicates whether the basal or apical side of the epithelium is closer to the constraint wall. Then, we investigated the relationship between the degree of asymmetry and epithelial folding morphology using three-dimensional vertex simulations. The results show that the folding patterns of the epithelial sheets change from spot patterns to labyrinth patterns and then to hole patterns as the degree of asymmetry changes. Furthermore, we examined the pattern formation in terms of the equation of out-of-plane displacement of the sheet derived from the mechanical energy functional.

Laser MBE growth of cobalt-platinum multilayers with perpendicular magnetic anisotropy

Periodic multilayers with perpendicular magnetic anisotropy (PMA) attract a lot of attention nowadays being fundamentally interesting from the point of view of interface magnetism. They also provide a suitable test bench for the studies of spatially resolved magnetization dynamics at picosecond time scale conducted at x-ray free electron laser (XFEL) facilities. In the present paper we have developed a novel laser molecular beam epitaxy (LMBE) approach to grow Co/Pt multilayers on crystalline Al2O3 (0001) substrates. By applying Kerr-effect magnetometry to the samples fabricated at different conditions, we have found it essential for the appearance of PMA, to perform metal deposition in 0.03 mbar of argon and to keep Co layer thickness below 4 Å. Along with magnetic studies, we have investigated crystal structure and depth resolved layer composition of the fabricated Co/Pt multilayers by applying electron diffraction, x-ray diffraction and x-ray reflectometry. The combined atomic-force/magnetic-force microscopy studies have revealed a very low surface roughness of down to 1 Å (for the samples grown at properly optimized growth conditions) and a pronounced labyrinth pattern of magnetic domains with a periodicity of 250 nm appropriate for XFEL studies of ultrafast optical magnetization. The presented LMBE approach to fabricate magnetically ordered heterostructures with a pronounced PMA is supposed to be an important milestone on the way of facilitating future design of nanomaterials for applications in magnetic memories and for fundamental studies of ultrafast magnetization dynamics.

Tunable Friction Properties of Periodic Wrinkled BaTiO3 Membranes

AbstractThe wrinkled patterns may find potential applications in the areas that require tunable friction properties, such as flexible electronics, nanomechanical systems, and biomedical devices. However, the regulation of wrinkled patterns and the influence of friction property remain poorly understood, especially in the field of oxides. The epitaxial wrinkled BaTiO3 (BTO) membranes are prepared by the pulsed laser deposition and water‐dissolution process. The patterns, amplitude, and wavelength of the wrinkled BTO membranes are dependent on the thickness, pre‐stretched mechanical stress, and heat treatment. The friction properties of the wrinkled BTO membranes exhibit various anisotropic and periodic features due to the varying wrinkled patterns as applying different mechanical stress, including anisotropy with 180° angular periodicity, anisotropy with 360° angular periodicity, and isotropy, corresponding to the parallel, zigzag, and labyrinth patterns, respectively. Moreover, the heat process can further alter the magnitude of the friction property, showing a downward trend with increasing temperature. These results provide a better understanding of the friction properties of wrinkled patterns and pave a novel way to alter the friction properties for extensive applications.

Study of periodic diffusion and time delay induced spatiotemporal patterns in a predator-prey system

In this paper, we establish a diffusion-reaction predator-prey model with Allee effect and time delay. For this model, we mainly study the effects of time delay and periodic diffusion on the stability of the equilibrium point and the formation of Turing pattern. First, we theoretically analyze the conditions of Turing instability caused by the time delay and periodic diffusion. At the same time, through a large number of numerical simulations, we verify the existence of spatiotemporal patterns and obtain that the pattern in the formation process not only appears the spots, strips, but also exists the coexistence of spots and strips. In addition, it also appears the labyrinth pattern under some suitable parameters. The final comparisons show that periodic diffusion, time delay and Allee effect all play an important role in the formation of Turing pattern.

Labyrinth patterns in Magadi (Kenya) cherts: Evidence for early formation from siliceous gels

Abstract Sedimentary cherts, with well-preserved microfossils, are known from the Archean to the present, yet their origins remain poorly understood. Lake Magadi, Kenya, has been used as a modern analog system for understanding the origins of nonbiogenic chert. We present evidence for synsedimentary formation of Magadi cherts directly from siliceous gels. Petrographic thin-section analysis and field-emission scanning electron microscopy of cherts from cores drilled in Lake Magadi during the Hominin Sites and Paleolakes Drilling Project in 2014 led to the discovery of two-dimensional branching “labyrinth patterns” in chert, which are a type of fractal “squeeze” pattern formed at air-liquid interfaces. Labyrinth patterns preserved in chert from Lake Magadi cores indicate invasion of air along planes in dewatering gels. These patterns support the precipitation of silica gels in the saline-alkaline Lake Magadi system and syndepositional drying of gels in contact with air as part of chert formation. Recognizing cherts as syndepositional has been critical for our use of them for U-Th dating. Identification of labyrinth patterns in ancient cherts can provide a better understanding of paleoenvironmental and geochemical conditions in the past.